Marder M.P. Condensed Matter Physics

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Thermoelectric Phenomena 503

40 60

Angle

<f>

(degrees)

100

Figure 17.5. Magnetoresistance of crystalline silver at T = 4.2 K with a magnetic field

of 2.35 T pointing along [001] . On the y axis is the change Ap in the resistivity divided

by the zero-field resistivity. Peaks correspond to open orbits. [Source: Alekseevskii and

Gaidukhov(1960), p. 673.]

17.4.10 Anomalous Hall Effect

In materials with spontaneous magnetic moments M, no external field is needed

in order to produce a Hall current. An electric field along i leads to an electrical

current pointing along y. This phenomenon is the anomalous Hall effect.

For many years it was debated whether the effect was intrinsic and present in

perfect crystals, as first proposed by Karplus and Luttinger (1954), or whether it

required impurities as proposed by Smit (1955). The resolution of the debate is

that both mechanisms appear simultaneously in practical experiments, but intrinsic

contributions to the anomalous Hall conductivity are indeed possible.

The intrinsic part of the anomalous Hall effect comes from the anomalous ve-

locity in Eq. (16.84). With a uniform non-zero electric field E and no magnetic

field, the electron velocity is

d£r eE -

—I + — X fir.

dhk n

(17.111)

The first term on the right hand side of

Eq.

(17.111) leads to currents parallel to Ë.

The second term however produces a contribution

; =

-Ex

J[dk]

Vï.

Since Ë is already a small quantity, there is

no need to keep track also of the difference

between g

^ and

/j.

Sum over multiple bands

if necessary.

(17.112)

and an anomalous Hall conductivity when j is along x and Ë is along y of

„2

&AH =

~h~

J[dk]

Cli f-. The integrand must also be summed over all

k k occupied bands.

(17.113)

It is not immediately obvious that Eq. (17.113) describes an anomalous Hall

effect that is proportional to the strength of spontaneous magnetization M. In

504 Chapter 17. Transport Phenomena and Fermi Liquid Theory

1000

7 500

0 12 3

(/VMn)

Figure 17.6. Comparison of experimental and theoretical values for the anomalous Hall

conductivity a

in ferromagnetic thin films of Mn5Ge3. The variation of temperature

from 2 to 400 K causes the spontaneous magnetic moment M

to vary, and causes a

vary as well. Contributions from impurity scattering known as side jump scattering have

been removed from the data. The theory appears to be a simple straight line, but in fact

results from elaborate band structure calculations without adjustable parameters. [Source:

Zeng et al. (2006), p. 3.]

a centrosymmetric crystal, the anomalous velocity f^ vanishes (Section 16.4.2);

spontaneous magnetization breaks the symmetry and allows the Berry connection

and anomalous velocity to develop. Demonstrating this idea in a quantitative way

requires the full machinery of band structure calculations.

Computation of Q^ leading to a prediction for anomalous Hall currents was

first carried out by Jungwirth et al. (2002). Figure 17.6 shows a comparison of

theory and experiment for the anomalous Hall effect in thin films of

Mn5Ge3.

The

spontaneous magnetization M

varies strongly as a function of temperature, and

the spontaneous Hall coefficient varies with it nearly linearly. Additional values of

computed and measured anomalous Hall coefficients appear in Table 17.2.

Table 17.2. Anomalous Hall conductivity at

low temperature for selected ferromàgnets.

bcc Fe fee Ni hep Co

Computation (S/cm) 753 -2203 447

Experiment (S/cm) 1032 -646 408

[Source: Wang et

al.

(2007), p. 7.]

17.5 Fermi Liquid Theory

17.5.1 Basic Ideas

Fermi liquid theory was developed by Landau (1956) for the purpose of explain-

ing

He, but it has a significance that reaches beyond this particular application.

It explains why a system of strongly interacting particles might continue to act in

Fermi Liquid Theory

505

accord with the single-particle approximation, and it provides a way to quantify

the changes produced by interactions. The basic idea is that one should focus on

excitations of the strongly interacting system, without worrying about the precise

nature of the ground state. These elementary excitations act like particles, and

therefore are also called quasi-particles. Their energies are nearly additive, and

one can build complicated excited states by adding together many quasi-particles.

Quasi-particles do interact with each other, but less strongly than the original par-

ticles from which they are constructed.

(A) (B)

Figure 17.7. (A) An excited state of

collection of

fermions,

where precisely one fermion

is given an energy above

£.

(B) In order for the fermion sitting outside the Fermi sea

to interact with those inside it, it must create a

final

state in which some particle has been

ejected from below 8.f to above Ef.

Here is a thought experiment to show where the quasi-particles come from.

Build a metal, or ajar of

He with a knob on one side. When the knob is at zero,

all of the interparticle interactions of the system are turned off. For a metal, the

Coulomb interaction between electrons is set to zero, and for helium the short-

range repulsion between helium atoms vanishes. When the knob is at 1, the in-

teractions between particles reach full strength. Placing the knob at zero creates

a noninteracting Fermi gas. Consider this gas in its ground state, except that one

particle is given wave vector k at energy £] a tiny bit above the Fermi surface, as

shown in Figure 17.7(A). All the excited states of the noninteracting system can be

described by exciting one or more particles above the Fermi surface in this way.

Now imagine turning the knob slowly to 1, so that all the particles below the

Fermi surface, in the Fermi sea, begin to interact with each other and with the

particle sitting above. According to the adiabatic theorem [see Landau and Lif-

shitz (1977), p. 148, or Schiff (1968), p. 289], if the knob turns slowly enough,

the system evolves continuously into an eigenstate of the interacting Hamiltonian.

However, this is not the situation of interest to Fermi liquid theory. Instead, to con-

struct Fermi liquid theory, imagine turning the knob at a rate that is rapid compared

to the scattering time r of k states near the Fermi surface. The result of turning the

knob will be a state that still has eigenvalue k, which means that it has eigenvalue

expD'À:

• R]

when translated through R. However, it will not be an energy eigenstate,

and will decay in time.

The essential point is that for states lying very close to the Fermi surface, the

scattering time r goes to infinity as (£i

—

£F)~

, which means that one can turn

the knob arbitrarily slowly and still end up with k states after the knob reaches 1.

506

Chapter 17. Transport Phenomena and

Fermi

Liquid Theory

It is possible to explain the divergence of

by examining the structure of the

electron scattering problem, without performing

detailed calculation. As shown

in Appendix C, when particles interact with two-body interactions, the interaction

Hamiltonian takes the form

[nt

= V

Ih,-ret

_ Â _ Cr Cr, ,.

(17.114)

Z^ k'qk k'-q,<r' k+q,a k,a k',a'

k'qk

According to Fermi's Golden Rule, Eq. (13.99), the rate at which a particle leaves

state

and goes to some other state is proportional to the number

final states

available after the scattering process. Possible final states are determined by ask-

ing

for

what states

the matrix element (^^f/intl*

)

nonzero. Only one

particle, with wave number k, lies above the Fermi surface in the initial state, so k'

in Eq. (17.114) must lie below the Fermi surface. If q

0 or q = k'

—

k, the final

state is the same as the initial state, and the wave number

the particle has not

changed. The only other possibility is for the final state to consist of two particles

above the Fermi surface, with wave vectors K

—

q and k

q, and

hole

wave

vector

below it, as shown in Figure 17.7(B). As the particle with wave number k

descends closer and closer to the Fermi surface, it becomes more and more difficult

to find acceptable final states. Equation (13.99) becomes

T(^^)

/(n^/0(£/))Y5(£

+£

-£3-£4)|(*

|^nt|*

. (17.115)

Taking into account the fact that £3 and £4 must be greater than

£f,

while £2

£F,

and taking the matrix element to be constant when all energies lie very close to the

Fermi surface, Eq. (17.115) becomes

y(k^k')(x

[

d£

[ '

rf£

oc(£i-£

)

ocr-

. (17.116)

J2£.

-£.\

JEF

As claimed, the scattering time of k states diverges near the Fermi surface.

To summarize: There exists a set of

wave

functions that are in one-to-one corre-

spondence with low-lying excited states of the noninteracting Fermi gas and behave

under translations like noninteracting particles with index k. One can also construct

states in correspondence with low-energy holes of the noninteracting gas. These

states are not true energy eigenfunctions, but they decay very slowly near the Fermi

surface.

17.5.2 Statistical Mechanics of Quasi-Particles

Energy Functional. Landau (1956) proposed a phenomenological description of

a quantum state inhabited by many quasi-particles. Let /^ describe the occupation

number of state k. In the ground state all f^ below the Fermi surface are

1 ;

all above

are 0. Next, let 8f^ describe the difference between the occupation of state k and its

Fermi Liquid Theory 507

occupation in the ground state; in the ground state, all 8

are 0. Suppose that the

energy of the quantum state can be expanded in terms of the occupation numbers

8fa (the spin state a will not be shown explicitly on 8fa unless necessary):

ka *.*'.

The function £i

gives the energy of noninteracting particles. For a metal, it is the

energy of Bloch wave functions, while for helium it is h

/2m. The function u^,

describes interactions between the quasi-particles.

Zero Temperature. The energy needed to add one quasi-particle 8fa above the

Fermi surface is

l'a

If the chemical potential ji rises, the system will fill with quasi-particles until the

cost of adding them rises above /i. Similarly, if the chemical potential drops, states

will empty out until £^ = /z. Therefore, in equilibrium at zero temperature where

ß =

8.F,

the occupation numbers / are

/j<

= 0(£f-£

). (17.119)

For an isotropic system the Fermi wave vector is defined by £^ = £f. Because

indexing of the quasi-particles is in one-to-one correspondence with free particles,

relationships for free particles are preserved, such as

*=E/*=*

= ^ = 3^-

(

°)

kcr

Low Temperatures. The grand partition function of the Fermi liquid is

= ]T exp { -/3[£(£f -ß)8n

l +

\ Ç 8

,8n^\ (17.121)

Srir —5rtj trr kk'

The integers 8n^ take values 0 and

above the Fermi surface, and

— 1

and 0 below

it, and are distinct from the occupation probabilities 8fa, which are the thermal

averages of the ôn's. This partition function cannot be summed exactly, but at

low temperatures its properties can be obtained with excellent accuracy from a

mean field approximation. To carry out the mean-field approximation, replace 8n

everywhere by

8n~

= S f~ + (8n of-) To learn why this approximation is called mean (

7 122)

k Jk K k Jk

> field theory, see Section 24.4.

508

Chapter 17.

Transport

Phenomena and

Fermi

Liquid Theory

and keep only up to linear order in the second term of (17.122), regarded as small.

The grand partition function becomes

gr =

Stir

=0,1...

^E*

Î -/

+E*'

Vk'

h'\

«' v*

ist'

ß5f-

,6f

-ί[£?-p]önj

iW;

««,*/*,

TJ(j

-ß[£

-iAh

(17.123)

(17.124)

(17.125)

**'

No-

where

hj.

— 1

if state k lies below the Fermi surface, and

otherwise.

The expectation value <5/j of occupying state £ is given by

*/Ï=n

gl W*

"M"

/*'

CTO"'

<5nj =0,1.

e-ft

i'->4

v/Zg,, (17.126)

Fa'

and because most terms cancel between numerator and denominator it is

Vï

—

fi '. f

defined

in Eq.

(17.119).

(^-

)

+ l e

ß(e

-ß)

+ l

(17.127)

So the quasi-particle states are occupied with a probability that is given by the

Fermi function. However, because £^ depends upon the occupation probabilities

through Eq. (17.118), Eq. (17.127) is a complicated implicit expression.

17.5.3 Effective Mass

The effective mass m* of quasi-particles is defined by

Hkf

(17.128)

Because of the interactions w-g,, the effective mass and bare particle mass differ.

The method employed to calculate the change is a generalization of the technique

used in Section 16.2.2. The idea is to compute the total particle current of a system

with quasi-particles in two different ways and, by comparing them, deduce the

effective mass.

First Calculation of Particle Current. The quasi-particle states were specially

chosen so as to be eigenfunctions of momentum, so evaluating the total

flow

particles 7/v is easy. For helium it is

Ä=E(*I-I*

(17.129)

—f-, = \

—Of?. The

difference

between

and

spheri-

(17.130)

"tr"*

cally

symmetrical.

Fermi Liquid Theory 509

For a metal, the matrix element (17.129) would be evaluated as following Eq. (16.24),

and it would give an effective mass due to interactions with the lattice, rather than

simply m.

Second Calculation of Particle Current. However, the current carried by a col-

lection of particles can also be calculated by calculating how their energy changes

when all their momenta change by a small amount.

To see why, consider the operator that generates small changes in momentum

P-

i i

Acting on a general Hamiltonian "K with this operator gives

[i-t£p-Rt/h] I Yl £;

\ E tfinttö.Rß) I [i+'Ep-RiW

(

132

)

= Ä + Vf-.

(17.133)

^ m

So particle current can be calculated by taking a collection of quasi-particles, in-

creasing all their momenta by p = hdk, finding the change in their mean energy,

and dividing through by

Hdk,

just as in Eq. (16.24).

To apply this idea to the energy function of

Eq.

(17.117), note that if

the

index k

of all quasi-particles increases by dk, then the occupation probability fa is replaced

by fa_

i- Increasing the momenta of

all

quasi-particles by hdk changes their energy

EtfVi-A-f?\+\ YVi-*-f?\*&\fv-A-$\

(

134

)

era'

d£- 1

=<**■£

fi-w+T,

f^+2 E Hi

«**/*■

135

)

la »,

era'

See Eq. (17.127). Multiply out the interaction term, whenever l

—

dl appears, change the

summation index sending J

—> &

+ dk, use u^, =

«J,J,

and expand out to first order in dl.

Differentiating the change in energy by hdk gives

•^=Ev*

(

136

)

with

lh = Derivatives of £r give the group velocity. M 7 137)

dhk

To express Eq. (17.136) in terms of of rather than /, write

A-L^ + E^-»«* <'7,38,

<J<J'

510

Chapter

17.

Transport Phenomena

and

Fermi Liquid Theory

<9£

(0)

-sA-Vi

ElVi+Zflss

"r^h

-'»>

OTT

8F~ dfi

= y^

èSfr

—

'S^ ^r

Urr,Sfr, Regroup integrals, then integrate (17.140)

Ôhk

dhk

kk Jk

"y parts.

era'

**'.

era

The second term

on the

right

Eq. (17.141)

interpreted

backflow,

a re-

verse flow

of the

medium around each particle, which inevitably accompanies

whenever one tries

drive

the

particle

create

current.

Comparing

the

Two Expressions. Comparing Eqs. (17.130)

and

(17.141)

in the

case where just

one Sf^ is

nonzero provides

now an

expression

for the

effective

mass induced

interactions,

^•

+ E

^^(E^-EF).

(17.142)

—

k'o'

hk'

+ V

Wrr,

5(£??-£

) Using definition (17.128). (17.143)

^-^ m*

k'a'

-fj

■?

Take

dot

product

both sides with

-, . S~^

srcW

c "\

Assuming

the

Fermi surface spherical,

A A

—

+ /

Tit—T

o(ci —tf) ,

, , . . ,

(17.144)

Z-^i kk' fol v if

both

k and

k! must have magnitude

because Fermi liquid theory

only

valid near the Fermi surface.

„

he angular integral di-,

dk'Dr,

S(oi°

- Of)

■

performed over

the

Fermi

(

17.145)

K' ' KK

surface.

r\i

The

density

states normally includes

the

-V-^ «77, COS

*"P

ular

intégral,

and

because that integral

is (

146

)

4TJ-

kk'

being performed separately, D(

)

divided

v 7

47T. Ö

is the

angle between

k and k!.

/■

only depends

on the

angle

= \+VD(£

d(cOs6)

«77, COS 6>.

between* and*',

and

could

(17.147)

2 J—

written

as a

function

of 6.

the

effective mass

increases over

the

bare mass

according

to a

weighted

average

the interactions

u-^,

over

the

Fermi surface.

17.5.4 Specific Heat

The specific heat

low temperatures

to the

density

states

at the

Fermi

surface D(£.p).

calculate

the

specific heat, note that

kk'

aa'

Ï-»F-

149

)

Fermi Liquid Theory 511

Using Eq. (17.127) for 5f^ gives

as/-

h^ik-v)

dôfp

dß

ÖT [«VM + lp

jtjr

T ^

Ukk

' ÖT k

ÖT

(17.150)

At low temperatures, the first term in curly brackets is much larger than the others

in Eq. (17.150), so

= V [dk] ~(£r-/u)

—^ r Everything is even in (17.151)

[ J

*' [eß(£i-ß) + l]2 % -n, so

goes away.

= V

J£ D(£)

(£

^[^-,)

l]2

(17

152)

VT j , . The integral over £ is performed as in Eq. (6.67). This . _ _

^ Cy——k

l U[tf). equation is identical to Eq. (6.77). (1/.153)

Because the density of states at the Fermi surface is also given by

D(£

)= f[dk]ö(£

-Er)= i dl

£f

Jj_ =

!??!**

(17.154)

the specific heat can be used to measure the effective mass.

17.5.5 Fermi Liquid Parameters

Moments of

U-Q,

over the Fermi surface provide the most important information

about interactions. There is a conventional notation for these moments, the Landau

parameters. This notation takes into account the possibility that particles with the

same spin have an interaction that differs from the interaction between particles

with different spin, so

"ïît'î =

hl'i =

U' +

(17

155

M<i =

k\k<\

ïï>

~ "8" (17.156)

where the s stands for "symmetric," and the a stands for "antisymmetric." Then

one defines

Ik'

'^2

/^/(

cos

/is a Legendre polynomial. (17.157)

/=0

h = Y. «Wcos0). (17.158)

1=0

The formulae may be inverted:

"? = ^ /' *(«* 0)fl(cos

ö)

JMl±^Mi

(

i7.i59)

2 7-i 2

/ d(cos 8)Pi(cos 0) "

\ "^^ (17.160)

7-i 2

2/+1

512

Chapter 17.

Transport

Phenomena and

Fermi

Liquid Theory

The parameters u all have dimensions of energy. To make them dimensionless,

multiply by the density VD(8.

) of energy states at the Fermi surface, to get

VD(E

)uf,

Ff = VD(E

) u\. (17.161)

Example: Effective Mass Relation. The integral appearing in Eq. (17.147) is

i r

VD(£

/ d(cos9) cos6u

, (17.162)

I) (J) f d(cos 9) A (cos e)VD(£

)

Uïfk

'î

Uîfk,i

(17.163)

Ff. (17.164)

So the effective mass relation (17.147) takes the form

— =

+ -Ff. (17.165)

m 3

17.5.6 Traveling Waves

The critical observer may complain that Fermi liquid theory is vacuous. It be-

gins with some unknown quantities, such as specific heat and effective mass, and

in attempting to improve matters it defines an infinite number of other unknown

quantities—the interaction parameters—and then relates new unknowns to the old

unknowns. Nothing has really been learned unless there are some cross-checks.

Landau provided a check by calculating the speeds of two different sorts of

traveling waves in

He. The first of them is ordinary sound, which is a traveling

pressure wave where the liquid remains in local thermal equilibrium as pressure

oscillates up and down. The second is zero

sound,

a traveling wave moving at

frequencies so high that thermal equilibrium has no time to establish

itself.

Ordinary First Sound. The speed of sound c in a liquid or gas is given by

2 =

=— \s

■

(17.166)

In liquid

He, the interest is in temperatures below 1 K. The difference between

isothermal and adiabatic derivatives is proportional to dS/dn\

, which vanishes

at T = 0 because S does. Therefore one can replace the adiabatic derivative in

Eq. (17.166) by an isothermal one and write

PIP -V ft,,

(17.167)

(17.168)

The motion of sound involves sloshing of mass back and forth; there should be no doubt that the

mass density p = rr/N/V involves the bare helium mass, not an effective mass. This argument

would not, however, work for a metal, where sound also involves motion of the ions.

VdP

mdN

N dp,

m dN

V d dF

~~m~dN~dV~

mdN

-Vdp

m dV